Fisher's inequality is a
necessary condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth o ...
for the existence of a balanced incomplete
block design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...
, that is, a system of subsets that satisfy certain prescribed conditions in
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. Outlined by
Ronald Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who ...
, a
population geneticist
Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and popu ...
and
statistician
A statistician is a person who works with theoretical or applied statistics. The profession exists in both the private and public sectors.
It is common to combine statistical knowledge with expertise in other subjects, and statisticians may w ...
, who was concerned with the
design of experiments
The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...
such as studying the differences among several different
varieties
Variety may refer to:
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* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
of plants, under each of a number of different growing conditions, called ''blocks''.
Let:
* be the number of varieties of plants;
* be the number of blocks.
To be a balanced incomplete block design it is required that:
* different varieties are in each block, ; no variety occurs twice in any one block;
* any two varieties occur together in exactly blocks;
* each variety occurs in exactly blocks.
Fisher's inequality states simply that
:: .
Proof
Let the incidence matrix be a matrix defined so that is 1 if element is in block and 0 otherwise. Then is a matrix such that and for . Since , , so ; on the other hand, , so .
Generalization
Fisher's inequality is valid for more general classes of designs. A ''pairwise balanced design'' (or PBD) is a set together with a family of non-empty subsets of (which need not have the same size and may contain repeats) such that every pair of distinct elements of is contained in exactly (a positive integer) subsets. The set is allowed to be one of the subsets, and if all the subsets are copies of , the PBD is called "trivial". The size of is and the number of subsets in the family (counted with multiplicity) is .
Theorem: For any non-trivial PBD, .
This result also generalizes the
Erdős–De Bruijn theorem:
For a PBD with having no blocks of size 1 or size , , with equality if and only if the PBD is a
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
or a near-pencil (meaning that exactly of the points are
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
).
In another direction,
Ray-Chaudhuri and
Wilson
Wilson may refer to:
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** List of people with given name Wilson
** List of people with surname Wilson
* Wilson (footballer, 1927–1998), Brazilian manager and defender
* Wilson (footballer, born 1984), full name Wilson R ...
proved in 1975 that in a design, the number of blocks is at least
.
Notes
References
*
R. C. Bose
Raj Chandra Bose (19 June 1901 – 31 October 1987) was an Indian American mathematician and statistician best known for his work in design theory, finite geometry and the theory of error-correcting codes in which the class of BCH codes is p ...
, "A Note on Fisher's Inequality for Balanced Incomplete Block Designs", ''
Annals of Mathematical Statistics
The ''Annals of Mathematical Statistics'' was a peer-reviewed statistics journal published by the Institute of Mathematical Statistics from 1930 to 1972. It was superseded by the ''Annals of Statistics'' and the ''Annals of Probability''. In 1938 ...
'', 1949, pages 619–620.
* R. A. Fisher, "An examination of the different possible solutions of a problem in incomplete blocks", ''
Annals of Eugenics
Annals ( la, annāles, from , "year") are a concise historical record in which events are arranged chronologically, year by year, although the term is also used loosely for any historical record.
Scope
The nature of the distinction between ann ...
'', volume 10, 1940, pages 52–75.
*
*
{{Experimental design, state=expanded
Combinatorial design
Design of experiments
Families of sets
Statistical inequalities